
The branching fraction of \(D_s^-\rightarrow \phi (\mu ^+\mu ^-) \pi ^-\)  is determined from known branching fractions taken from Ref. [1]}.
The branching fraction of \(D_s^-\rightarrow \tau ^- \nu _{\overline{\tau }}\)  is also taken from Ref. [1]}.
The quantity \(f_{\tau }^{D_{s}}\)  is the fraction of \(\tau ^-\)  that come from \(D_s^-\)  decays, calculated using the \(b\overline{b}\)  and \(c\overline{c}\) 
cross-sections as measured by LHCb [3]}, [4]}, the inclusive \(b\rightarrow \tau ^-\)  and \(c\rightarrow \tau ^-\) 
rates as measured by the LEP experiments and various branching fractions from Ref. [1]}.
This term is required as \(D_s^-\rightarrow \tau ^- \nu _{\overline{\tau }}\)  only accounts for \(\sim 80\%\)  of the production of \(\tau ^-\)  leptons.
The total efficiencies, \(\epsilon _{\tau \rightarrow X\mu \mu }\)  and \(\epsilon _{D_{s}\rightarrow \phi \pi }\) , take
into account the acceptance, selection and trigger efficiencies for the signal and normalisation channels,
whilst \(N_{D_{s}\rightarrow \phi \pi }\)  is the number of reconstructed \(D_s^-\rightarrow \phi (\mu ^+\mu ^-) \pi ^-\)  events in data, and \(N_{\tau \rightarrow X\mu \mu }\)  is the observed number of signal events.
